The equation of hyperbola $H$ is $\dfrac {(y+3)^{2}}{36}-\dfrac {(x+5)^{2}}{64} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+3)^{2}}{36} = 1 + \dfrac {(x+5)^{2}}{64}$ Multiply both sides of the equation by $36$ $(y+3)^{2} = { 36 + \dfrac{ (x+5)^{2} \cdot 36 }{64}}$ Take the square root of both sides. $\sqrt{(y+3)^{2}} = \pm \sqrt { 36 + \dfrac{ (x+5)^{2} \cdot 36 }{64}}$ $ y + 3 = \pm \sqrt { 36 + \dfrac{ (x+5)^{2} \cdot 36 }{64}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 3 \approx \pm \sqrt {\dfrac{ (x+5)^{2} \cdot 36 }{64}}$ $y + 3 \approx \pm \left(\dfrac{6 \cdot (x + 5)}{8}\right)$ Subtract $3$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{3}{4}(x + 5) -3$